Related papers: On maximal multiplicities for Hamiltonians with se…

In this note, we compute the higher spin spectrum of $U(M)_k$ Chern-Simons theory coupled to $N$ flavours of fundamental fermions, in the limit $N\gg M$ with the 't Hooft coupling $\lambda_M =\frac{N}{k_m}$ held fixed, to order $M/N$. This…

Let $A$ be a nonempty finite subset of an additive abelian group $G$. Given a nonnegative integer $h$, the $h$-fold sumset $hA$ is the set of all sums of $h$ elements of $A$, and the restricted $h$-fold sumset $h^\wedge A$ is the set of all…

An $n$-list $\lambda:=\left(r; \lambda_2, \ldots, \lambda_n\right)$ of complex numbers with $r>0,$ is said to be realizable if $\lambda$ is the spectrum of $n\times n$ nonnegative matrix $A$ and in this case $A$ is said to be a nonnegative…

For $n \geq 2$ consider the affine Lie algebra $\widehat{s\ell}(n)$ with simple roots $\{\alpha_i \mid 0 \leq i \leq n-1\}$. Let $V(k\Lambda_0), \, k \in \mathbb{Z}_{\geq 1}$ denote the integrable highest weight $\widehat{s\ell}(n)$-module…

In the high dimensional Stochastic Blockmodel for a random network, the number of clusters (or blocks) K grows with the number of nodes N. Two previous studies have examined the statistical estimation performance of spectral clustering and…

A non-Hermitean random matrix model proposed a few years ago has a remarkably intricate spectrum. Various attempts have been made to understand the spectrum, but even its dimension is not known. Using the Dyson-Schmidt equation, we show…

Let $(\mathfrak{g}, \mathfrak{k})$ be a complex quaternionic symmetric pair with $\mathfrak{k}$ having an ideal $\mathfrak{sl}(2, \mathbb{C})$, $\mathfrak{k}=\mathfrak{sl}(2, \mathbb{C})+\mathfrak{m}_c$. Consider the representation…

We establish uniform bounds on the multiplicities of irreducible admissible representations appearing in spaces of functions on symmetric spaces over $p$-adic fields. These multiplicities can exceed one and depend intricately on the group,…

Fix integers $b>a\geq1$ with $g:=\gcd(a,b)$. A set $S\subseteq\mathbb{N}$ is \emph{$\{a,b\}$-multiplicative} if $ax\neq by$ for all $x,y\in S$. For all $n$, we determine an $\{a,b\}$-multiplicative set with maximum cardinality in $[n]$, and…

We study the minimizers of \begin{equation} \lambda_k^s(A) + |A| \end{equation} where $\lambda^s_k(A)$ is the $k$-th Dirichlet eigenvalue of the fractional Laplacian on $A$. Unlike in the case of the Laplacian, the free boundary of…

For $\lambda \in \mathbb{Z}$, let $\lambda \cdot A = \{ \lambda a : a \in A\}$. Suppose $r, h\in \mathbb{Z}$ are sufficiently large and comparable to each other. We prove that if $|A+A| \le K |A|$ and $\lambda_1, \ldots, \lambda_h \le 2^r$,…

We classify $n$-representation infinite algebras $\Lambda$ of type \~A. This type is defined by requiring that $\Lambda$ has higher preprojective algebra $\Pi_{n+1}(\Lambda) \simeq k[x_1, \ldots, x_{n+1}] \ast G$, where $G \leq…

Let $Q_n$ be the $n$-dimensional Hamming cube and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically \[2n2^{N/4},\] as was conjectured by Ilinca and the first author in connection with a question of…

We introduce and carefully study a natural probability measure over the numerical range of a complex matrix $A \in M_n(\C)$. This numerical measure $\mu_A$ can be defined as the law of the random variable $<AX,X> \in \C$ when the vector $X…

Suppose that c is a linear operator acting on an n-dimensional complex Hilbert Space H, and let tau denote the normalized trace on B(H). Set b_1 = (c+c*)/2 and b_2 = (c-c*)/2i, and write B for the the spectral scale of {b_1, b_2} with…

An explicit Dirichlet series is obtained, which represents an analytic function of $s$ in the half-plane $\Re s>1/2$ except for having simple poles at points $s_j$ that correspond to exceptional eigenvalues $\lambda_j$ of the non-Euclidean…

Consider a sparse system of n Laurent polynomials in n variables with complex coefficients and support in a finite lattice set A. The maximal number of isolated roots of the system in the complex n-torus is known to be the normalized volume…

We define eventually symmetric functions to be those power series of bounded degree in infinitely many variables that are invariant under interchanging all the variables with large enough indices. We show how this ring $\tilde{\Lambda}$ is…

In an analytically unramified local ring $(R,\mathfrak m)$ of dimension $d\geq 1$, for a filtration of ideals $\mathfrak {I}=\{I_m\}_{m\in\mathbb N}$ satisfying $\mathfrak A(r)$ condition and for any $\mathfrak m$-primary ideal $K$, it is…

We prove the density of the sets of the form ${{\lambda}_1^m {\mu}_1^n {\xi}_1 +...+{\lambda}_k^m {\mu}_k^n {\xi}_k : m,n \in \mathbb N}$ modulo one, where $\lambda_i$ and $\mu_i$ are multiplicatively independent algebraic numbers…